Boolean and Free Symmetrization of Bernoulli Distributions

TL;DR

The paper addresses variance bounds under symmetry across classical, free, and Boolean probability, using Bernoulli distributions and their noncommutative analogues (projections with trace $p$). It proves a sharp lower bound: in the classical case, if $X\sim\mathrm{Bernoulli}(p)$ with $p\neq \tfrac{1}{2}$ and $X+Y$ is symmetric about $0$, then $\mathrm{Var}(Y)\ge pq$; in the free and Boolean settings, for a projection $e$ with $\phi(e)=p$ and a self-adjoint $y$ free or Boolean independent of $e$ with $e+y$ symmetric, one has $\phi(y^2)\ge p$, with equality for $y\stackrel{d}{=}-e$. The work also unveils symmetry phenomena specific to Boolean convolution, including symmetry-non-resistance (symmetry arising from non-symmetric measures) and non-uniqueness of symmetrizers in both Boolean and free contexts. It connects these variance bounds to quantum-information contexts, notably a two-level system example, and outlines directions for extending the theory to critical cases, other noncommutative independences, and higher-dimensional settings. Overall, the results unify variance inequalities across probabilistic frameworks and illuminate how symmetry enforces rigidity in noncommutative stochastic models with potential applications in quantum information and noncommutative probability theory.

Abstract

We investigate variance bounds under symmetry constraints in classical, free, and Boolean probability, focusing on Bernoulli distributions and their noncommutative analogues, projections with trace \(p\). We show that symmetrizers under classical, free, and Boolean convolution satisfy a sharp variance bound of \(pq\), with equality for the reflection law. Additionally, we highlight phenomena specific to Boolean convolution, demonstrating that non-symmetric measures can produce symmetric convolutions and that symmetrizers may be non-unique for certain measures. These results unify variance inequalities across probabilistic frameworks and offer insights for quantum information and noncommutative stochastic modeling.

Theorems & Definitions (15)

• Theorem 1: Classical case
• Theorem 2: Free case
• Theorem 3: Boolean case
• Remark 1
• Proposition 1
• Proposition 2
• Conjecture 1
• Proposition 3
• Proposition 4
• proof : Proof of Theorem \ref{['thm:free']}